Idea

Lie integration is a process that assigns to a Lie algebra𝔤\mathfrak{g} – or more generally to an ∞-Lie algebra or ∞-Lie algebroid – a Lie group – or more generally ∞-Lie groupoid – that is infinitesimally modeled by 𝔤\mathfrak{g}. It is essentially the reverse operation to Lie differentiation, except that there are in general several objects Lie integrating a given Lie algebraic datum, due to the fact that the infinitesimal data does not uniquely determine global topological properties.

Notice that this induces some degree shifts that may be a little ambiguous in situations like the line Lie n-algebra: as an L-∞ algebra this is bn−1ℝb^{n-1}\mathbb{R}, the corresponding ∞-Lie algebroid is bnℝb^n \mathbb{R}.

Remark

The Lie integration of 𝔞\mathfrak{a} is essentially the simplicial object whose kk-cells are the dd-paths in 𝔞\mathfrak{a}. However, in order for this to be well-behaved, it is possible and useful to restrict to dd-paths that are sufficiently well-behaved towards the boundary of the simplex:

Definition

Regard the smooth simplex Δk\Delta^k as embedded into the Cartesian spaceℝk+1\mathbb{R}^{k+1} in the standard way, and equip Δk\Delta^k with the metric space structure induced this way.

A smooth differential formω\omega on Δk\Delta^k is said to have sitting instants along the boundary if, for every (r<k)(r \lt k)-face FF of Δk\Delta^k there is an open neighbourhoodUFU_F of FF in Δk\Delta^k such that ω\omega restricted to UU is constant in the directions perpendicular to the rr-face on its value restricted to that face.

More generally, for any U∈U \in CartSp a smooth differential form ω\omega on U×ΔkU \times\Delta^k is said to have sitting instants if there is 0<ϵ∈ℝ0 \lt \epsilon \in \mathbb{R} such that for all points u:*→Uu : * \to U the pullback along (u,Id):Δk→U×Δk(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k is a form with sitting instants on ϵ\epsilon-neighbourhoods of faces.

Smooth forms with sitting instants clearly form a sub-dg-algebra of all smooth forms. We write Ωsi•(U×Δk)\Omega^\bullet_{si}(U \times \Delta^k) for this sub-dg-algebra.

We write Ωsi,vert•(U×Δk)\Omega_{si,vert}^\bullet(U \times \Delta^k) for the further sub-dg-algebra of vertical differential forms with respect to the projection p:U×Δk→Up : U \times \Delta^k \to U, hence the coequalizer

Then the pullback form ϕ*ω∈Ω•(Δk)\phi^* \omega \in \Omega^\bullet(\Delta^k) is a form with sitting instants.

Remark

The condition of sitting instants serves to make smooth differential forms not be affected by the boundaries and corners of Δk\Delta^k. Notably for ωj∈Ω•(Δk−1)\omega_j \in \Omega^\bullet(\Delta^{k-1}) a collection of forms with sitting instants on the (k−1)(k-1)-cells of a hornΛik\Lambda^k_i that coincide on adjacent boundaries, and for

glue to a single smooth differential form (with sitting instants) on Δk\Delta^k.

Remark

That ω∈Ω•(Δk)\omega \in \Omega^\bullet(\Delta^k) having sitting instants does not imply that there is a neighbourhood of the boundary of Δk\Delta^k on which ω\omega is entirely constant. It is important for the following constructions that in the vicinity of the boundary ω\omega is allowed to vary parallel to the boundary, just not perpendicular to it.

Remark

To see this, consider the example (discussed in detail below) that 𝔞=𝔤\mathfrak{a} = \mathfrak{g} is an ordinary Lie algebra. Then exp(𝔤)n\exp(\mathfrak{g})_n is canonically identified with the set of smooth based maps Δn→G\Delta^n \to G into the simply connected Lie group that integrates 𝔤\mathfrak{g} in ordinary Lie theory. This means that the simplicial homotopy groups of exp(𝔤)\exp(\mathfrak{g}) are the topological homotopy groups of GG, which in general (say for GG the orthogonal group or unitary group) will be non-trivial in arbitrarily higher degree, even though 𝔤\mathfrak{g} is just a Lie 1-algebra. This phenomenon is well familiar from rational homotopy theory, where a classical theorem asserts that the rational homotopy groups of exp(𝔤)\exp(\mathfrak{g}) are generated from the generators in a minimal Sullivan model resolution of 𝔤\mathfrak{g}.

Proposition

Proof

Observe that the standard continuoushornretractsf:Δk→Λikf : \Delta^k \to \Lambda^k_i are smooth away from the preimages of the (r<k)(r \lt k)-faces of Λ[k]i\Lambda[k]^i.

For ω∈Ωsi,vert•(U×Λ[k]i)\omega \in \Omega^\bullet_{si,vert}(U \times \Lambda[k]^i) a differential form with sitting instants on ϵ\epsilon-neighbourhoods, let therefore K⊂∂ΔkK \subset \partial \Delta^k be the set of points of distance ≤ϵ\leq \epsilon from any subface. Then we have a smooth function

Proof

The bijection is given as follows. For A∈Ωflat1(X,𝔤)A \in \Omega^1_{flat}(X,\mathfrak{g}) a flat 1-form, the corresponding function fA:X→Gf_A : X \to G sends x∈Xx \in X to the parallel transport along any path x0→xx_0 \to x from the base point to xx

fA:x↦traA(x0→x).
f_A : x \mapsto tra_A(x_0 \to x)
\,.

Because of the assumption that the curvature 2-form of AA vanishes and the assumption that XX is simply connected, this assignment is independent of the choice of path.

Conversely, for every such function f:X→Gf : X \to G we recover AA as the pullback of the Maurer-Cartan form on GG

The ∞\infty-groupoid cosk2exp(𝔤)\mathbf{cosk}_2 \exp(\mathfrak{g}) is equivalent to the groupoid with a single object (no non-trivial 1-form on the point) whose morphisms are equivalence classes of smooth based paths Δ1→G\Delta^1 \to G (with sitting instants), where two of these are taken to be equivalent if there is a smooth homotopyD2→GD^2 \to G (with sitting instant) between them.

Since GG is simply connected, these equivalence classes are labeled by the endpoints of these paths, hence are canonically identified with GG.

Remark

We do not need to fall back to classical Lie theory to obtain GG in the above argument. A detailed discussion of how to find GG with its group structure and smooth structure from dd-paths in 𝔤\mathfrak{g} is in (Crainic).

Observation

The discrete ∞-groupoid underlying exp(bn−1ℝ)\exp(b^{n-1} \mathbb{R}) is given by the Kan complex that in degree kk has the set of closed differential nn-forms (with sitting instants) on the kk-simplex

Proposition

The ∞\infty-Lie integration of bn−1ℝb^{n-1} \mathbb{R} is the circle n-groupBnℝ\mathbf{B}^{n} \mathbb{R}.

Moreover, with Bnℝchn∈[CartSpsmoothop,sSet]\mathbf{B}^n \mathbb{R}_{chn} \in [CartSp_{smooth}^{op}, sSet] the standard presentation given under the Dold-Kan correspondence by the chain complex of sheaves concentrated in degree nn on C∞(−,ℝ)C^\infty(-, \mathbb{R}) the equivalence is induced by the fiber integration of differential nn-forms over the nn-simplex:

The only nontrivial degree to check is degree nn. Let λ∈Ωsi,vert,cln(Δn+1)\lambda \in \Omega_{si,vert,cl}^n(\Delta^{n+1}). The differential of the normalized chains complex sends this to the signed sum of its restrictions to the nn-faces of the (n+1)(n+1)-simplex. Followed by the integral over Δn\Delta^n this is the piecewise integral of λ\lambda over the boundary of the nn-simplex. Since λ\lambda has sitting instants, there is 0<ϵ∈ℝ0 \lt \epsilon \in \mathbb{R} such that there are no contributions to this integral in an ϵ\epsilon-neighbourhood of the (n−1)(n-1)-faces. Accordingly the integral is equivalently that over the smooth surface inscribed into the (n+1)(n+1)-simplex, as indicated in the following diagram

Since λ\lambda is a closed form on the nn-simplex, this surface integral vanishes, by the Stokes theorem. Hence ∫Δ•\int_{\Delta^\bullet} is indeed a chain map.

It remains to show that ∫Δ•:exp(bn−1ℝ)→Bnℝchn\int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn} is an isomorphism on all the simplicial homotopys group over each U∈CartSpU \in CartSp. This amounts to the statement that

a smooth family of closed nn-forms with sitting instants on the boundary of Δn+1\Delta^{n+1} may be extended to a smooth family of closed forms with sitting instants on Δn+1\Delta^{n+1} precisely if their smooth family of integrals over the boundary vanishes;

Any smooth family of closed n<kn \lt k-forms with sitting instants on the boundary of Δk+1\Delta^{k+1} may be extended to a smooth family of closed nn-forms with sitting instants on Δk+1\Delta^{k+1}.

To demonstrate this, we want to work with forms on the (k+1)(k+1)-ball instead of the (k+1)(k+1)-simplex. To achieve this, choose again 0<ϵ∈ℝ0 \lt \epsilon \in \mathbb{R} and construct the diffeomorphic image of Sk×[1,1−ϵ]S^k \times [1,1-\epsilon] inside the (k+1)(k+1)-simplex as indicated in the above diagram: outside an ϵ\epsilon-neighbourhood of the corners the image is a rectangular ϵ\epsilon-thickening of the faces of the simplex. Inside the ϵ\epsilon-neighbourhoods of the corners it bends smoothly. By the Steenrod-Wockel approximation theorem the diffeomorphism from this ϵ\epsilon-thickening of the smoothed boundary of the simplex to Sk×[1−ϵ,1]S^k \times [1-\epsilon,1] extends to a smooth function from the (k+1)(k+1)-simplex to the (k+1)(k+1)-ball.

By choosing ϵ\epsilon smaller than each of the sitting instants of the given nn-form on ∂Δk+1\partial \Delta^{k+1}, we have that this nn-form vanishes on the ϵ\epsilon-neighbourhoods of the corners and is hence entirely determined by its restriction to the smoothed simplex, identified with the (k+1)(k+1)-ball.

It is now sufficient to show: a smooth family of smooth nn-forms ω∈Ωvert,cln(U×Sk)\omega \in \Omega^n_{vert,cl}(U \times S^k) extends to a smooth family of closed nn-forms ω^∈Ωvert,cln(U×Bk+1)\hat \omega \in \Omega^n_{vert,cl}(U \times B^{k+1}) that is radially constant in a neighbourhood of the boundary for all n<kn \lt k and for k=nk = n precisely if its smooth family of integrals vanishes, ∫Skω=0∈C∞(U,ℝ)\int_{S^k} \omega = 0 \in C^\infty(U, \mathbb{R}).

Notice that over the point this is a direct consequence of the de Rham theorem: an nn-form ω\omega on SkS^k is exact precisely if n<kn \lt k or if n=kn = k and its integral vanishes. In that case there is an (n−1)(n-1)-form AA with ω=dA\omega = d A. Choosing any smoothing function f:[0,1]→[0,1]f : [0,1] \to [0,1] (smooth, surjective, non,decreasing and constant in a neighbourhood of the boundary) we obtain an nn-form f∧Af \wedge A on (0,1]×Sk(0,1] \times S^k, vertically constant in a neighbourhood of the ends of the interval, equal to AA at the top and vanishing at the bottom. Pushed forward along the canonical (0,1]×Sk→Dk+1(0,1] \times S^k \to D^{k+1} this defines a form on the (k+1)(k+1)-ball, that we denote by the same symbol f∧Af \wedge A. Then the form ω^:=d(f∧A)\hat \omega := d (f \wedge A) solves the problem.

To complete the proof we have to show that this simple argument does extend to smooth families of forms, i.e., that we can choose the (n−1)(n-1)-form AA in a way depending smoothly on the the nn-form ω\omega.

Integrating the string Lie 2-algebra to the string Lie 2-group

Then cosk3exp(𝔤μ)\mathbf{cosk}_3 \exp(\mathfrak{g}_\mu) is equivalent to the 2-groupoidBString\mathbf{B}String

with a single object;

whose morphisms are based paths in GG;

whose 2-morphisms are equivalence class of pairs (Σ,c)(\Sigma,c), where

Σ:D*2→G\Sigma : D^2_* \to G is a smooth based map (where we use a homeomorphismD2≃Δ2D^2 \simeq \Delta^2 which away from the corners is smooth, so that forms with sitting instants there do not see any non-smoothness, and the basepoint of D*2D^2_* is the 0-vertex of Δ2\Delta^2)

and c∈U(1)c \in U(1), and where two such are equivalent if the maps coincides at their boundary and if for any 3-ball ϕ:D3→G\phi : D^3 \to G filling them the labels c1,c2∈U(1)c_1, c_2 \in U(1) differ by the integral ∫D3ϕ*μ(θ)modℤ\int_{D^3} \phi^* \mu(\theta) \;\; mod \;\; \mathbb{Z},,

where θ\theta is the Maurer-Cartan form, μ(θ)=⟨θ∧[θ∧θ]⟩\mu(\theta) = \langle \theta\wedge [\theta \wedge \theta]\rangle the 3-form obtained by plugging it into the cocycle.